Editing Nest algebra (section)

In [[functional analysis]], a branch of mathematics, '''nest algebras''' are a class of [[operator algebra]]s that generalise the [[upper-triangular matrix]] algebras to a [[Hilbert space]] context. They were introduced by {{harvs|txt|authorlink=John Ringrose|last=Ringrose|year=1965}} and have many interesting properties. They are non-[[selfadjoint]] algebras, are [[closure (mathematics)|closed]] in the [[weak operator topology]] and are [[reflexive operator algebra|reflexive]].

Nest algebras are among the simplest examples of [[commutative subspace lattice algebra]]s. Indeed, they are formally defined as the algebra of [[bounded operator]]s leaving [[Invariant (mathematics)|invariant]] each [[Linear subspace|subspace]] contained in a [[subspace nest]], that is, a set of subspaces which is [[total order|totally ordered]] by [[subset|inclusion]] and is also a [[complete lattice]]. Since the [[orthogonal projection]]s corresponding to the subspaces in a nest [[commutative operation|commute]], nests are commutative subspace lattices.

By way of an example, let us apply this definition to recover the finite-dimensional upper-triangular matrices. Let us work in the <math>n</math>-[[dimension]]al [[complex number|complex]] [[vector space]] <math>\mathbb{C}^n</math>, and let <math>e_1,e_2,\dots,e_n</math> be the [[standard basis]]. For <math>j=0,1,2,\dots,n</math>, let <math>S_j</math> be the <math>j</math>-dimensional subspace of <math>\mathbb{C}^n</math> [[linear span|span]]ned by the first <math>j</math> basis vectors <math>e_1,\dots,e_j</math>. Let

:<math>N=\{ (0)=S_0, S_1, S_2, \dots, S_{n-1}, S_n=\mathbb{C}^n \};</math>

then ''N'' is a subspace nest, and the corresponding nest algebra of ''n''&nbsp;&times;&nbsp;''n'' complex matrices ''M'' leaving each subspace in ''N'' invariant &nbsp; that is, satisfying <math>MS\subseteq S</math> for each ''S'' in ''N'' &ndash; is precisely the set of upper-triangular matrices.

If we omit one or more of the subspaces ''S<sub>j</sub>'' from ''N'' then the corresponding nest algebra consists of block upper-triangular matrices.